20
JOHN AITCHISON
In the limit, as
dt
---* 0,
the differential perturbation process
{14.2)
is equivalent to
the set of differential equations
{14.4)
dt
d
log
Xi((t))
=
8i(t)- 8;(t) (i j).
Xj
t
There is redundancy in this set of!
D(D -1)
differential equations; the reduced set
{14.5)
d Xi(t)
-d log-(-)
=
8i(t)- 8n(t),
i
=
1, ... ,d,
t xn t
is sufficient to describe the process. Given the initial composition
x(O)
at time
0
we can readily obtain the composition
x(t)
at timet for the process by solving the
differential equations
{14.6)
x(t)
=
x(O)
o
P(t),
where the
ith
component
~(t)
of the overall perturbation
P(t)
is
(14.7)
~(t)
= exp
{lot
8i(u)du}.
We note that
{14.6)
and
{14.7)
can be expressed in terms of log ratios:
{14.8)
log{xi(t)/xn(t)}
=
log{xi(O)/xn(O)}
+
8i(t)- 8n(t),
i
=
1, ... , d,
and so we may anticipate that log ratio analysis of compositional data sets will
have a central role to play in the investigation of differential perturbation processes.
For purposes of exposition we have restricted our use of differential perturbation
processes to the case of a single continuous process variable
t.
No such restric-
tions are necessary. There may be a number of subprocesses, each with its own
driving continuous variable, in which case the differential equations defining the
process will involve partial derivatives. For compositional data sets with no contin-
uous variable defined, a log contrast principal component analysis will determine
the dimensionality of any such underlying continuous vector. Investigation of the
constant log contrast principal components is then the first step in any attempt to
identify the underlying processes. For some simple applications to sedimentary and
olivine compositions, see
[A-T].
15. Discussion
The history of simplicial inference abounds with all sorts of devices purport-
ing to overcome the difficulties associated with the special nature of the simplex
sample space. These range from imagining that there is no problem, ignoring the
constraint and inappropriately applying standard unconstrained
RD
multivariate
analysis; 'opening out' the compositions into imaginary vectors in
R!;_;
recognizing
that corr(xi,x;) = 0 does not mean that components
Xi
and
x;
are independent
but attempting to discover a non-zero 'null correlation' which characterizes inde-
pendence in imagined open vectors; studying compositional pathology in the sense
of identifying what goes
~rong
in applying standard multivariate analysis to sim-
plicial data; transforming from the simplex to the positive orthant of the sphere
and imagining that distributions over the whole sphere are appropriate for further
analysis; attempting to rid the Dirichlet class of its inherent independence proper-
ties. See
[Al2]
for further discussion of these attempted solutions.
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